Tipler_Llewellyn.pdf

Publisher: Clancy Marshall Senior Acquisitions Editor: Jessica Fiorillo Marketing Manager: Anthony Palmiotto Media Editors: Jeanette Picerno and Samantha Calamari Supplements Editor and Editorial Assistant: Janie Chan Senior Project Editor: Mary Louise Byrd Cover and Text Designer: Diana Blume Phot...

Publisher: Clancy Marshall Senior Acquisitions Editor: Jessica Fiorillo Marketing Manager: Anthony Palmiotto Media Editors: Jeanette Picerno and Samantha Calamari Supplements Editor and Editorial Assistant: Janie Chan Senior Project Editor: Mary Louise Byrd Cover and Text Designer: Diana Blume Photo Editor: Ted Szczepanski Photo Researcher: Rae Grant Senior Illustration Coordinator: Bill Page Production Coordinator: Paul W. Rohloff Illustrations and Composition: Preparé Printing and Binding: Quebecor Printing Library of Congress Control Number: 2007931523 ISBN-13: 978-0-7167-7550-8 ISBN-10: 0-7167-7550-6 © 2008 by Paul A. Tipler and Ralph A. Llewellyn All rights reserved. Printed in the United States of America First printing W. H. Freeman and Company 41 Madison Avenue New York, NY 10010 Houndmills, Basingstoke RG21 6XS, England www.whfreeman.com MODERN PHYSICS Fifth Edition Paul A. Tipler Formerly of Oakland University Ralph A. Llewellyn University of Central Florida W. H. Freeman and Company New York This page intentionally left blank Contents PART 1 Relativity and Quantum Mechanics: The Foundations of Modern Physics 1 CHAPTER 1 Relativity I 3 1-1 The Experimental Basis of Relativity 4 Michelson-Morley Experiment 11 1-2 Einstein’s Postulates 11 1-3 The Lorentz Transformation 17 Calibrating the Spacetime Axes 28 1-4 Time Dilation and Length Contraction 29 1-5 The Doppler Effect 41 Transverse Doppler Effect 44 1-6 The Twin Paradox and Other Surprises 45 The Case of the Identically Accelerated Twins 48 Superluminal Speeds 52 CHAPTER 2 Relativity II 65 2-1 Relativistic Momentum 66 2-2 Relativistic Energy 70 From Mechanics, Another Surprise 80 2-3 Mass/Energy Conversion and Binding Energy 81 2-4 Invariant Mass 84 The indicates material that appears only on the Web site: www.whfreeman.com/tiplermodernphysics5e. The indicates material of high interest to students. iv Contents 2-5 General Relativity 97 Deflection of Light in a Gravitational Field 100 Gravitational Redshift 103 Perihelion of Mercury’s Orbit 105 Delay of Light in a Gravitational Field 105 CHAPTER 3 Quantization of Charge, Light, and Energy 115 3-1 Quantization of Electric Charge 115 3-2 Blackbody Radiation 119 3-3 The Photoelectric Effect 127 3-4 X Rays and the Compton Effect 133 Derivation of Compton’s Equation 138 CHAPTER 4 The Nuclear Atom 147 4-1 Atomic Spectra 148 4-2 Rutherford’s Nuclear Model 150 Rutherford’s Prediction and Geiger and Marsden’s Results 156 4-3 The Bohr Model of the Hydrogen Atom 159 Giant Atoms 168 4-4 X-Ray Spectra 169 4-5 The Franck-Hertz Experiment 174 A Critique of Bohr Theory and the “Old Quantum Mechanics” 176 CHAPTER 5 The Wavelike Properties of Particles 185 5-1 The de Broglie Hypothesis 185 5-2 Measurements of Particle Wavelengths 187 5-3 Wave Packets 196 5-4 The Probabilistic Interpretation of the Wave Function 202 5-5 The Uncertainty Principle 205 The Gamma-Ray Microscope 206 5-6 Some Consequences of the Uncertainty Principle 208  Contents v 5-7 Wave-Particle Duality 212 Two-Slit Interference Pattern 213 CHAPTER 6 The Schrödinger Equation 221 6-1 The Schrödinger Equation in One Dimension 222 6-2 The Infinite Square Well 229 6-3 The Finite Square Well 238 Graphical Solution of the Finite Square Well 241 6-4 Expectation Values and Operators 242 Transitions Between Energy States 246 6-5 The Simple Harmonic Oscillator 246 Schrödinger’s Trick 249 Parity 250 6-6 Reflection and Transmission of Waves 250 Alpha Decay 258 NH3 Atomic Clock 260 Tunnel Diode 260 CHAPTER 7 Atomic Physics 269 7-1 The Schrödinger Equation in Three Dimensions 269 7-2 Quantization of Angular Momentum and Energy in the Hydrogen Atom 272 7-3 The Hydrogen Atom Wave Functions 281 7-4 Electron Spin 285 Stern-Gerlach Experiment 288 7-5 Total Angular Momentum and the Spin-Orbit Effect 291 7-6 The Schrödinger Equation for Two (or More) Particles 295 7-7 Ground States of Atoms: The Periodic Table 297 7-8 Excited States and Spectra of Atoms 301 Multielectron Atoms 303 The Zeeman Effect 303 Frozen Light 304 vi Contents CHAPTER 8 Statistical Physics 315 8-1 Classical Statistics: A Review 316 Temperature and Entropy 319 A Derivation of the Equipartition Theorem 324 8-2 Quantum Statistics 328 8-3 The Bose-Einstein Condensation 335 Liquid Helium 336 8-4 The Photon Gas: An Application of Bose-Einstein Statistics 344 8-5 Properties of a Fermion Gas 351 PART 2 Applications of Quantum Mechanics and Relativity 361 CHAPTER 9 Molecular Structure and Spectra 363 9-1 The Ionic Bond 364 9-2 The Covalent Bond 369 Other Covalent Bonds 375 9-3 Other Bonding Mechanisms 375 9-4 Energy Levels and Spectra of Diatomic Molecules 379 9-5 Scattering, Absorption, and Stimulated Emission 390 9-6 Lasers and Masers 396 CHAPTER 10 Solid State Physics 413 10-1 The Structure of Solids 413 10-2 Classical Theory of Conduction 422 10-3 Free-Electron Gas in Metals 426 10-4 Quantum Theory of Conduction 430 Thermal Conduction—The Quantum Model 434 10-5 Magnetism in Solids 434 Spintronics 437 10-6 Band Theory of Solids 438 Energy Bands in Solids—An Alternate Approach 445  Contents vii 10-7 Impurity Semiconductors 445 Hall Effect 449 10-8 Semiconductor Junctions and Devices 452 How Transistors Work 457 10-9 Superconductivity 458 Flux Quantization 462 Josephson Junction 466 CHAPTER 11 Nuclear Physics 477 11-1 The Composition of the Nucleus 478 11-2 Ground-State Properties of Nuclei 480 Liquid-Drop Model and the Semiempirical Mass Formula 489 11-3 Radioactivity 492 Production and Sequential Decays 495 11-4 Alpha, Beta, and Gamma Decay 495 Energetics of Alpha Decay 498 The Mössbauer Effect 505 11-5 The Nuclear Force 506 Probability Density of the Exchange Mesons 512 11-6 The Shell Model 513 Finding the “Correct” Shell Model 516 11-7 Nuclear Reactions 516 11-8 Fission and Fusion 526 Nuclear Power 530 Interaction of Particles and Matter 536 11-9 Applications 537 Radiation Dosage 549 CHAPTER 12 Particle Physics 561 12-1 Basic Concepts 562 12-2 Fundamental Interactions and the Force Carriers 570 A Further Comment About Interaction Strengths 577 viii Contents 12-3 Conservation Laws and Symmetries 580 When Is a Physical Quantity Conserved? 583 Resonances and Excited States 591 12-4 The Standard Model 591 Where Does the Proton Get Its Spin? 595 12-5 Beyond the Standard Model 605 Neutrino Oscillations and Mass 609 Theories of Everything 610 CHAPTER 13 Astrophysics and Cosmology 619 13-1 The Sun 619 Is There Life Elsewhere? 630 13-2 The Stars 630 The Celestial Sphere 636 13-3 The Evolution of Stars 639 13-4 Cataclysmic Events 644 13-5 Final States of Stars 647 13-6 Galaxies 653 13-7 Cosmology and Gravitation 662 13-8 Cosmology and the Evolution of the Universe 664 “Natural” Planck Units 673 Appendix A Table of Atomic Masses AP-1 Appendix B Mathematical Aids AP-16 B1 Probability Integrals AP-16 B2 Binomial and Exponential Series AP-18 B3 Diagrams of Crystal Unit Cells AP-19 Appendix C Electron Configurations AP-20 Appendix D Fundamental Physical Constants AP-26 Appendix E Conversion Factors AP-30 Appendix F Nobel Laureates in Physics AP-31 Answers AN-1 Index I-1 Preface I n preparing this new edition of Modern Physics, we have again relied heavily on the many helpful suggestions from a large team of reviewers and from a host of instruc- tor and student users of the earlier editions. Their advice reflected the discoveries that have further enlarged modern physics in the early years of this new century and took note of the evolution that is occurring in the teaching of physics in colleges and uni- versities. As the term modern physics has come to mean the physics of the modern era—relativity and quantum theory—we have heeded the advice of many users and reviewers and preserved the historical and cultural flavor of the book while being careful to maintain the mathematical level of the fourth edition. We continue to pro- vide the flexibility for instructors to match the book and its supporting ancillaries to a wide variety of teaching modes, including both one- and two-semester courses and media-enhanced courses. Features The successful features of the fourth edition have been retained, including the following: The logical structure—beginning with an introduction to relativity and quantiza- tion and following with applications—has been continued. Opening the book with relativity has been endorsed by many reviewers and instructors. As in the earlier editions, the end-of-chapter problems are separated into three sets based on difficulty, with the least difficult also grouped by chapter section. More than 10 percent of the problems in the fifth edition are new. The first edition’s Instructor’s Solutions Manual (ISM) with solutions, not just answers, to all end-of- chapter problems was the first such aid to accompany a physics (and not just a modern physics) textbook, and that leadership has been continued in this edition. The ISM is available in print or on CD for those adopting Modern Physics, fifth edition, for their classes. As with the previous edition, a paperback Student’s Solution Manual containing one-quarter of the solutions in the ISM is also available. We have continued to include many examples in every chapter, a feature singled out by many instructors as a strength of the book. As before, we frequently use combined quantities such as hc, Uc, and ke2 in eV # nm to simplify many numerical calculations. The summaries and reference lists at the end of every chapter have, of course, been retained and augmented, including the two-column format of the summaries, which improves their clarity. ix x Preface We have continued the use of real data in figures, photos of real people and appa- ratus, and short quotations from many scientists who were key participants in the development of modern physics. These features, along with the Notes at the end of each chapter, bring to life many events in the history of science and help counter the too-prevalent view among students that physics is a dull, impersonal collection of facts and formulas. More than two dozen Exploring sections, identified by an atom icon and dealing with text-related topics that captivate student interest such as superluminal speed and giant atoms, are distributed throughout the text. The book’s Web site includes 30 MORE sections, which expand in depth on many text-related topics. These have been enthusiastically endorsed by both students and instructors and often serve as springboards for projects and alternate credit assign- ments. Identified by a laptop icon , each is introduced with a brief text box. More than 125 questions intended to foster discussion and review of concepts are distributed throughout the book. These have received numerous positive comments from many instructors over the years, often citing how the questions encourage deeper thought about the topic. Continued in the new edition are the Application Notes. These brief notes in the margins of many pages point to a few of the many benefits to society that have been made possible by a discovery or development in modern physics. New Features A number of new features are introduced in the fifth edition: The “Astrophysics and Cosmology” chapter that was on the fourth edition’s Web site has been extensively rewritten and moved into the book as a new Chapter 13. Emphasis has been placed on presenting scientists’ current understanding of the evolution of the cosmos based on the research in this dynamic field. The “Particle Physics” chapter has been substantially reorganized and rewritten focused on the remarkably successful Standard Model. As the new Chapter 12, it immediately precedes the new “Astrophysics and Cosmology” chapter to recog- nize the growing links between these active areas of current physics research. The two chapters concerned with the theory and applications of nuclear physics have been integrated into a new Chapter 11, “Nuclear Physics.” Because of the renewed interest in nuclear power, that material in the fourth edition has been aug- mented and moved to a MORE section of the Web. Recognizing the need for students on occasion to be able to quickly review key concepts from classical physics that relate to topics developed in modern physics, we have added a new Classical Concept Review (CCR) to the book’s Web site. Identified by a laptop icon in the margin near the pertinent modern physics topic of discussion, the CCR can be printed out to provide a convenient study sup- port booklet. The Instructor’s Resource CD for the fifth edition contains all the illustrations from the book in both PowerPoint and JPEG format. Also included is a gallery of the astronomical images from Chapter 13 in the original image colors. Several new MORE sections have been added to the book’s Web site, and a few for which interest has waned have been removed.  Preface xi Organization and Coverage This edition, like the earlier ones, is divided into two parts: Part 1, “Relativity and Quantum Mechanics: The Foundation of Modern Physics,” and Part 2, “Applica- tions.” We continue to open Part 1 with the two relativity chapters. This location for relativity is firmly endorsed by users and reviewers. The rationale is that this arrangement avoids separation of the foundations of quantum mechanics in Chapters 3 through 8 from its applications in Chapters 9 through 12. The two-chap- ter format for relativity provides instructors with the flexibility to cover only the basic concepts or to go deeper into the subject. Chapter 1 covers the essentials of special relativity and includes discussions of several paradoxes, such as the twin paradox and the pole-in-the-barn paradox, that never fail to excite student interest. Relativistic energy and momentum are covered in Chapter 2, which concludes with a mostly qualitative section on general relativity that emphasizes experimental tests. Because the relation E 2  p2 c2  (mc2)2 is the result most needed for the later applications chapters, it is possible to omit Chapter 2 without disturbing conti- nuity. Chapters 1 through 8 have been updated with a number of improved explana- tions and new diagrams. Several classical foundation topics in those chapters have been moved to the Classical Concept Review or recast as MORE sections. Many quantitative topics are included as MORE sections on the Web site. Examples of these are the derivation of Compton’s equation (Chapter 3), the details of Ruther- ford’s alpha-scattering theory (Chapter 4), the graphical solution of the finite square well (Chapter 6), and the excited states and spectra of two-electron atoms (Chapter 7). The comparisons of classical and quantum statistics are illustrated with several examples in Chapter 8, and unlike the other chapters in Part 1, Chapter 8 is arranged to be covered briefly and qualitatively if desired. This chapter, like Chapter 2, is not essential to the understanding of the applications chapters of Part 2 and may be used as an applications chapter or omitted without loss of continuity. Preserving the approach used in the previous edition, in Part 2 the ideas and methods discussed in Part 1 are applied to the study of molecules, solids, nuclei, particles, and the cosmos. Chapter 9 (“Molecular Structure and Spectra”) is a broad, detailed discussion of molecular bonding and the basic types of lasers. Chapter 10 (“Solid-State Physics”) includes sections on bonding in metals, magnetism, and superconductivity. Chapter 11 (“Nuclear Physics”) is an integration of the nuclear theory and applications that formed two chapters in the fourth edition. It focuses on nuclear structure and properties, radioactivity, and the applications of nuclear reactions. Included in the last topic are fission, fusion, and several techniques of age dating and elemental analysis. The material on nuclear power has been moved to a MORE section, and the discussion of radiation dosage continues as a MORE section. As mentioned above, Chapter 12 (“Particle Physics”) has been substantially reorganized and rewritten with a focus on the Standard Model and revised to reflect the advances in that field since the earlier editions. The emphasis is on the funda- mental interactions of the quarks, leptons, and force carriers and includes discus- sions of the conservation laws, neutrino oscillations, and supersymmetry. Finally, the thoroughly revised Chapter 13 (“Astrophysics and Cosmology”) examines the current observations of stars and galaxies and qualitatively integrates our discus- sions of quantum mechanics, atoms, nuclei, particles, and relativity to explain our present understanding of the origin and evolution of the universe from the Big Bang to dark energy. xii Preface The Research Frontier Research over the past century has added abundantly to our understanding of our world, forged strong links from physics to virtually every other discipline, and measurably improved the tools and devices that enrich life. As was the case at the beginning of the last century, it is hard for us to foresee in the early years of this century how scientific research will deepen our understanding of the physical universe and enhance the quality of life. Here are just a few of the current subjects of frontier research included in Modern Physics, fifth edition, that you will hear more of in the years just ahead. Beyond these years there will be many other discoveries that no one has yet dreamed of. The Higgs boson, the harbinger of mass, may now be within our reach at Brookhaven’s Relativistic Heavy Ion Collider and at CERN with completion of the Large Hadron Collider. (Chapter 12) The neutrino mass question has been solved by the discovery of neutrino oscilla- tions at the Super-Kamiokande and SNO neutrino observatories (Chapters 2, 11, and 12), but the magnitudes of the masses and whether the neutrino is a Majorana particle remain unanswered. The origin of the proton’s spin, which may include contributions from virtual strange quarks, still remains uncertain. (Chapter 11) The Bose-Einstein condensates, which suggest atomic lasers and super–atomic clocks are in our future, were joined in 2003 by Fermi-Dirac condensates, wherein pairs of fermions act like bosons at very low temperatures. (Chapter 8) It is now clear that dark energy accounts for 74 percent of the mass> energy of the universe. Only 4 percent is baryonic (visible) matter. The remaining 22 percent consists of as yet unidentified dark matter particles. (Chapter 13) The predicted fundamental particles of supersymmetry (SUSY), an integral part of grand unification theories, will be a priority search at the Large Hadron Collider. (Chapters 12 and 13) High-temperature superconductors reached critical temperatures greater than 130 K a few years ago and doped fullerenes compete with cuprates for high-Tc records, but a theoretical explanation of the phenomenon is not yet in hand. (Chapter 10) Gravity waves from space may soon be detected by the upgraded Laser Interfero- metric Gravitational Observatory (LIGO) and several similar laboratories around the world. (Chapter 2) Adaptive-optics telescopes, large baseline arrays, and the Hubble telescope are providing new views deeper into space of the very young universe, revealing that the expansion is speeding up, a discovery supported by results from the Sloan Digital Sky Survey and the Wilkinson Microwave Anisotropy Project. (Chapter 13) Giant Rydberg atoms, made accessible by research on tunable dye lasers, are now of high interest and may provide the first direct test of the correspondence principle. (Chapter 4) The search for new elements has reached Z ⴝ 118, tantalizingly near the edge of the “island of stability.” (Chapter 11) Many more discoveries and developments just as exciting as these are to be found throughout Modern Physics, fifth edition.  Preface xiii Some Teaching Suggestions This book is designed to serve well in either one- or two-semester courses. The chap- ters in Part 2 are independent of one another and can be covered in any order. Some possible one-semester courses might consist of Part 1, Chapters 1, 3, 4, 5, 6, 7; and Part 2, Chapters 11, 12 Part 1, Chapters 3, 4, 5, 6, 7, 8; and Part 2, Chapters 9, 10 Part 1, Chapters 1, 2, 3, 4, 5, 6, 7; and Part 2, Chapter 9 Part 1, Chapters 1, 3, 4, 5, 6, 7; and Part 2, Chapters 11, 12, 13 Possible two-semester courses might consist of Part 1, Chapters 1, 3, 4, 5, 6, 7; and Part 2, Chapters 9, 10, 11, 12, 13 Part 1, Chapters 1, 2, 3, 4, 5, 6, 7, 8; and Part 2, Chapters 9, 10, 11, 12, 13 There is tremendous potential for individual student projects and alternate credit assignments based on the Exploring and, in particular, the MORE sections. The latter will encourage students to search for related sources on the Web. Acknowledgments Many people contributed to the success of the earlier editions of this book, and many more have helped with the development of the fifth edition. We owe our thanks to them all. Those who reviewed all or parts of this book, offering suggestions for the fifth edition, include Marco Battaglia Richard Gelderman University of California–Berkeley Western Kentucky University Mario Belloni Tim Gfroerer Davidson College Davidson College Eric D. Carlson Torgny Gustafsson Wake Forest University Rutgers University David Cinabro Scott Heinekamp Wayne State University Wells College Carlo Dallapiccola Adrian Hightower University of Massachusetts–Amherst Occidental College Anthony D. Dinsmore Mark Hollabaugh University of Massachusetts–Amherst Normandale Community College Ian T. Durham Richard D. Holland II Saint Anselm College Southern Illinois University at Carbondale Jason J. Engbrecht Bei-Lok Hu St. Olaf College University of Maryland–College Park Brian Fick Dave Kieda Michigan Technological University University of Utah Massimiliano Galeazzi Steve Kraemer University of Miami Catholic University of America Hugh Gallagher Wolfgang Lorenzon Tufts University University of Michigan xiv Preface Bryan A. Luther Ben E. K. Sugerman Concordia College at Moorhead Goucher College Catherine Mader Rein Uritam Hope College Physics Department Kingshuk Majumdar Boston College Berea College Ken Voss Peter Moeck University of Miami Portland State University Thad Walker Robert M. Morse University of Wisconsin–Madison University of Wisconsin–Madison Barry C. Walker Igor Ostrovskii University of Delaware University of Mississippi at Oxford Eric Wells Anne Reilly Augustana College College of William and Mary William R. Wharton David Reitze Wheaton College University of Florida Weldon J. Wilson Mark Riley University of Central Oklahoma Florida State University R. W. M. Woodside Nitin Samarth University College of Fraser Valley Pennsylvania State University Kate Scholberg Duke University We also thank the reviewers of the fourth and third editions. Their comments significantly influenced and shaped the fifth edition as well. For the fourth edition they were Darin Acosta, University of Florida; Jeeva Anandan, University of South Carolina; Gordon Aubrecht, Ohio State University; David A. Bahr, Bemidji State University; Patricia C. Boeshaar, Drew University; David P. Carico, California Polytechnic State University at San Luis Obispo; David Church, University of Washington; Wei Cui, Purdue University; Snezana Dalafave, College of New Jersey; Richard Gass, University of Cincinnati; David Gerdes, University of Michigan; Mark Hollabaugh, Normandale Community College; John L. Hubisz, North Carolina State University; Ronald E. Jodoin, Rochester Institute of Technology; Edward R. Kinney, University of Colorado at Boulder; Paul D. Lane, University of St. Thomas; Fernando J. Lopez-Lopez, Southwestern College; Dan MacIsaac, Northern Arizona University; Robert Pompi, SUNY at Binghamton; Warren Rogers, Westmont College; George Rutherford, Illinois State University; Nitin Samarth, Pennsylvania State University; Martin A. Sanzari, Fordham University; Earl E. Scime, West Virginia University; Gil Shapiro, University of California at Berkeley; Larry Solanch, Georgia College & State University; Francis M. Tam, Frostburg State University; Paul Tipton, University of Rochester; K. Thad Walker, University of Wisconsin at Madison; Edward A. Whittaker, Stevens Institute of Technology; Stephen Yerian, Xavier University; and Dean Zollman, Kansas State University. For the third edition, reviewers were Bill Bassichis, Texas A&M University; Brent Benson, Lehigh University; H. J. Biritz, Georgia Institute of Technology; Patrick Briggs, The Citadel; David A. Briodo, Boston College; Tony Buffa, California Polytechnic State University at San Luis Obispo; Duane Carmony, Purdue University; Ataur R. Chowdhury, University of Alaska at Fairbanks; Bill Fadner, University of Northern Colorado; Ron Gautreau, New Jersey Institute of Technology; Charles Glashauser,  Preface xv Rutgers–The State University of New Jersey; Roger Hanson, University of Northern Iowa; Gary G. Ihas, University of Florida; Yuichi Kubota, University of Minnesota; David Lamp, Texas Tech University; Philip Lippel, University of Texas at Arlington; A. E. Livingston, University of Notre Dame; Steve Meloma, Gustavus Adolphus College; Benedict Y. Oh, Pennsylvania State University; Paul Sokol, Pennsylvania State University; Thor F. Stromberg, New Mexico State University; Maurice Webb, University of Wisconsin at Madison; and Jesse Weil, University of Kentucky. All offered valuable suggestions for improvements, and we appreciate their help. In addition, we give a special thanks to all the physicists and students from around the world who took time to send us kind words about the third and fourth editions and offered suggestions for improvements. Finally, though certainly not least, we are grateful for the support, encouragement, and patience of our families throughout the project. We especially want to thank Mark Llewellyn for his preparation of the Instructor’s Solutions Manual and the Student’s Solutions Manual and for his numerous helpful suggestions from the very beginning of the project, Eric Llewellyn for his photographic and computer-generated images, David Jonsson at Uppsala University for his critical reading of every chapter of the fourth edition, and Jeanette Picerno for her imaginative work on the Web site. Finally, to the entire Modern Physics team at W. H. Freeman and Company goes our sincerest appreciation for their skill, hard work, understanding about deadlines, and support in bringing it all together. Paul A. Tipler, Ralph A. Llewellyn, Berkeley, California Oviedo, Florida This page intentionally left blank PART 1 Relativity and Quantum Mechanics: The Foundations of Modern Physics The earliest recorded systematic efforts to assemble knowledge about motion as a key to un- derstanding natural phenomena were those of the ancient Greeks. Set forth in sophisticated form by Aristotle, theirs was a natural philosophy (i.e., physics) of explanations deduced from assumptions rather than experimentation. For example, it was a fundamental assumption that every substance had a “natural place” in the universe. Motion then resulted when a substance was trying to reach its natural place. Time was given a similar absolute meaning, as moving from some instant in the past (the creation of the universe) toward some end goal in the future, its natural place. The remarkable agreement between the deductions of Aristotelian physics and motions observed throughout the physical universe, together with a nearly total absence of accurate instruments to make contradictory measurements, led to ac- ceptance of the Greek view for nearly 2000 years. Toward the end of that time a few scholars had begun to deliberately test some of the predictions of theory, but it was Italian scientist Galileo Galilei who, with his brilliant experiments on motion, established for all time the absolute necessity of experimentation in physics and, coincidentally, initiated the disintegra- tion of Aristotelian physics. Within 100 years Isaac Newton had generalized the results of Galileo’s experiments into his three spectacularly successful laws of motion, and the natural philosophy of Aristotle was gone. With the burgeoning of experimentation, the following 200 years saw a multitude of major discoveries and a concomitant development of physical theories to explain them. Most of the latter, then as now, failed to survive increasingly sophisticated experimental tests, but by the dawn of the twentieth century Newton’s theoretical explanation of the motion of mechanical systems had been joined by equally impressive laws of electromagnetism and thermodynamics as expressed by Maxwell, Carnot, and others. The remarkable success of these laws led many scientists to believe that description of the physical universe was com- plete. Indeed, A. A. Michelson, speaking to scientists near the end of the nineteenth century, said, “The grand underlying principles have been firmly established... the future truths of physics are to be looked for in the sixth place of decimals.” 1  Such optimism (or pessimism, depending on your point of view) turned out to be pre- mature, as there were already vexing cracks in the foundation of what we now refer to as classical physics. Two of these were described by Lord Kelvin, in his famous Baltimore Lectures in 1900, as the “two clouds” on the horizon of twentieth-century physics: the fail- ure of theory to account for the radiation spectrum emitted by a blackbody and the inex- plicable results of the Michelson-Morley experiment. Indeed, the breakdown of classical physics occurred in many different areas: the Michelson-Morley null result contradicted Newtonian relativity, the blackbody radiation spectrum contradicted predictions of thermo- dynamics, the photoelectric effect and the spectra of atoms could not be explained by elec- tromagnetic theory, and the exciting discoveries of x rays and radioactivity seemed to be outside the framework of classical physics entirely. The development of the theories of quan- tum mechanics and relativity in the early twentieth century not only dispelled Kelvin’s “dark clouds,” they provided answers to all of the puzzles listed here and many more. The ap- plications of these theories to such microscopic systems as atoms, molecules, nuclei, and fundamental particles and to macroscopic systems of solids, liquids, gases, and plasmas have given us a deep understanding of the intricate workings of nature and have revolu- tionized our way of life. In Part 1 we discuss the foundations of the physics of the modern era, relativity theory, and quantum mechanics. Chapter 1 examines the apparent conflict between Einstein’s prin- ciple of relativity and the observed constancy of the speed of light and shows how accepting the validity of both ideas led to the special theory of relativity. Chapter 2 discusses the relations connecting mass, energy, and momentum in special relativity and concludes with a brief dis- cussion of general relativity and some experimental tests of its predictions. In Chapters 3, 4, and 5 the development of quantum theory is traced from the earliest evidences of quantiza- tion to de Broglie’s hypothesis of electron waves. An elementary discussion of theSchrödinger equation is provided in Chapter 6, illustrated with applications to one-dimensional systems. Chapter 7 extends the application of quantum mechanics to many-particle systems and introduces the important new concepts of electron spin and the exclusion principle. Concluding the development, Chapter 8 discusses the wave mechanics of systems of large numbers of identical particles, underscoring the importance of the symmetry of wave func- tions. Beginning with Chapter 3, the chapters in Part 1 should be studied in sequence because each of Chapters 4 through 8 depends on the discussions, developments, and examples of the previous chapters. 2 CHAPTER 1 Relativity I T he relativistic character of the laws of physics began to be apparent very early in the evolution of classical physics. Even before the time of Galileo and Newton, Nicolaus Copernicus1 had shown that the complicated and imprecise 1-1 The Experimental Basis of Relativity 4 Aristotelian method of computing the motions of the planets, based on the assumption 1-2 Einstein’s that Earth was located at the center of the universe, could be made much simpler, Postulates 11 though no more accurate, if it were assumed that the planets move about the Sun 1-3 The Lorenz instead of Earth. Although Copernicus did not publish his work until very late in Transformation 17 life, it became widely known through correspondence with his contemporaries and helped pave the way for acceptance a century later of the heliocentric theory of 1-4 Time Dilation planetary motion. While the Copernican theory led to a dramatic revolution in human and Length thought, the aspect that concerns us here is that it did not consider the location of Contraction 29 Earth to be special or favored in any way. Thus, the laws of physics discovered 1-5 The Doppler on Earth could apply equally well with any point taken as the center — i.e., the Effect 41 same equations would be obtained regardless of the origin of coordinates. This 1-6 The Twin invariance of the equations that express the laws of physics is what we mean by the Paradox and term relativity. Other Surprises 45 We will begin this chapter by investigating briefly the relativity of Newton’s laws and then concentrate on the theory of relativity as developed by Albert Einstein (1879–1955). The theory of relativity consists of two rather different theories, the special theory and the general theory. The special theory, developed by Einstein and others in 1905, concerns the comparison of measurements made in different frames of reference moving with constant velocity relative to each other. Contrary to popu- lar opinion, the special theory is not difficult to understand. Its consequences, which can be derived with a minimum of mathematics, are applicable in a wide variety of situations in physics and engineering. On the other hand, the general theory, also developed by Einstein (around 1916), is concerned with accelerated reference frames and gravity. Although a thorough understanding of the general theory requires more sophisticated mathematics (e.g., tensor analysis), a number of its basic ideas and important predictions can be discussed at the level of this book. The general theory is of great importance in cosmology and in understanding events that occur in the 3 4 Chapter 1 Relativity I vicinity of very large masses (e.g., stars) but is rarely encountered in other areas of physics and engineering. We will devote this chapter entirely to the special theory (often referred to as special relativity) and discuss the general theory in the final section of Chapter 2, following the sections concerned with special relativistic mechanics. 1-1 The Experimental Basis of Relativity Classical Relativity In 1687, with the publication of the Philosophiae Naturalis Principia Mathematica, Newton became the first person to generalize the observations of Galileo and others into the laws of motion that occupied much of your attention in introductory physics. The second of Newton’s three laws is dv Fm  ma 1-1 dt where dv>dt  a is the acceleration of the mass m when acted upon by a net force F. Equation 1-1 also includes the first law, the law of inertia, by implication: if F  0, then dv>dt  0 also, i.e., a  0. (Recall that letters and symbols in boldface type are vectors.) As it turns out, Newton’s laws of motion only work correctly in inertial reference frames, that is, reference frames in which the law of inertia holds.2 They also have the remarkable property that they are invariant, or unchanged, in any reference frame that moves with constant velocity relative to an inertial frame. Thus, all inertial frames are equivalent — there is no special or favored inertial frame relative to which absolute measurements of space and time could be made. Two such inertial frames are illus- trated in Figure 1-1, arranged so that corresponding axes in S and S are parallel and S moves in the x direction at velocity v for an observer in S (or S moves in the x S S y y v x x z z Figure 1-1 Inertial reference frame S is attached to Earth (the palm tree) and S to the cyclist. The corresponding axes of the frames are parallel, and S moves at speed v in the x direction of S.  1-1 The Experimental Basis of Relativity 5 (a) y´ → → (b) y´ → → v=0 a=0 v>0 a=0 → v S´ O´ S´ O´ y y x´ x´ z´ z´ x x S O O z S z (c) y´ → → v>0 a>0 → v S´ O´ ϑ → y a x´ z´ x S O z Figure 1-2 A mass suspended by a cord from the roof of a railroad boxcar illustrates the relativity of Newton’s second law, F  ma. The only forces acting on the mass are its weight mg and the tension T in the cord. (a) The boxcar sits at rest in S. Since the velocity v and the acceleration a of the boxcar (i.e., the system S) are both zero, both observers see the mass hanging vertically at rest with F  F  0. (b) As S moves in the x direction with v constant, both observers see the mass hanging vertically but moving at v with respect to O in S and at rest with respect to the S observer. Thus, F  F  0. (c) As S moves in the x direction with a  0 with respect to S, the mass hangs at an angle   0 with respect to the vertical. However, it is still at rest (i.e., in equilibrium) with respect to the observer in S, who now “explains” the angle  by adding a pseudoforce Fp in the x direction to Newton’s second law. direction at velocity v for an observer in S). Figures 1-2 and 1-3 illustrate the con- ceptual differences between inertial and noninertial reference frames. Transformation of the position coordinates and the velocity components of S into those of S is the Galilean transformation, Equations 1-2 and 1-3, respectively. x  x  vt y  y z  z t  t 1-2 uxœ  ux  v uyœ  uy uzœ  uz 1-3 ω y´ ω S´ Satellite z´ x´ Figure 1-3 A geosynchronous satellite has an orbital angular velocity y equal to that of Earth and, therefore, is always located above a particular Earth Geosynchronous orbit point on Earth; i.e., it is at rest with respect to the surface of Earth. An S observer in S accounts for the radial, or centripetal, acceleration a of the x satellite as the result of the net force FG. For an observer O at rest on Earth (in S), however, a  0 and FG  ma. To explain the acceleration z being zero, observer O must add a pseudoforce Fp  FG. 6 Chapter 1 Relativity I Notice that differentiating Equation 1-3 yields the result a  a since dv>dt  0 for constant v. Thus, F  ma  F œ. This is the invariance referred to above. Generalizing this result: Any reference frame that moves at constant velocity with respect to an iner- tial frame is also an inertial frame. Newton’s laws of mechanics are invariant in all reference systems connected by a Galilean transformation. Speed of Light In about 1860 James Clerk Maxwell summarized the experimental observations of electricity and magnetism in a consistent set of four concise equations. Unlike Newton’s laws of motion, Maxwell’s equations are not invariant under a Galilean transformation between inertial reference frames (Figure 1-4). Since the Maxwell equations predict the existence of electromagnetic waves whose speed would be a par- ticular value, c  1> 1 0 P0  3.00 108 m>s, the excellent agreement between this number and the measured value of the speed of light3 and between the predicted po- larization properties of electromagnetic waves and those observed for light provided strong confirmation of the assumption that light was an electromagnetic wave and, therefore, traveled at speed c.4 That being the case, it was postulated in the nineteenth century that electromagnetic waves, like all other waves, propagated in a suitable material medium. The implication of this postulate was that the medium, called the ether, filled the entire universe, including the interior of matter. (The Greek philosopher Aristotle had first suggested that the universe was permeated with “ether” 2000 years earlier.) In this way the remarkable opportunity arose to establish experimentally the existence of the all-pervasive ether by measuring the speed of light c relative to Earth as Earth moved relative to the ether at Classical Concept Review speed v, as would be predicted by Equation 1-3. The value of c was given by the The concepts of classical Maxwell equations, and the speed of Earth relative to the ether, while not known, was relativity, frames of assumed to be at least equal to its orbital speed around the Sun, about 30 km> s. Since reference, and coordinate the maximum observable effect is of the order v 2>c2 and given this assumption transformations — all v 2>c2 艐 108, an experimental accuracy of about 1 part in 108 is necessary in order important background to to detect Earth’s motion relative to the ether. With a single exception, equipment and our discussions of special relativity — may not have been emphasized in many introductory courses. As an y y´ aid to a better S S´ understanding of the v q concepts of modern y1 physics, we have included x x´ the Classical Concept Review on the book’s Web site. As z z´ you proceed through Modern Physics, the icon Figure 1-4 The observers in S and S see identical electric fields 2k > y1 at a distance y1  y1œ in the margin will alert from an infinitely long wire carrying uniform charge per unit length. Observers in both S and you to potentially helpful S measure a force 2kq > y1 on q due to the line of charge; however, the S observer measures classical background an additional force  0 v2q>(2 y1) due to the magnetic field at y 1œ arising from the motion of pertinent to the adjacent the wire in the x direction. Thus, the electromagnetic force does not have the same form in topics. different inertial systems, implying that Maxwell’s equations are not invariant under a Galilean transformation.  1-1 The Experimental Basis of Relativity 7 techniques available at the time had an experimental accuracy of only about 1 part in 10 4, woefully insufficient to detect the predicted small effect. That single exception was the experiment of Michelson and Morley.5 Questions 1. What would the relative velocity of the inertial systems in Figure 1-4 need to be in order for the S observer to measure no net electromagnetic force on the charge q? 2. Discuss why the very large value for the speed of the electromagnetic waves would imply that the ether be rigid, i.e., have a large bulk modulus. The Michelson-Morley Experiment All waves that were known to nineteenth-century scientists required a medium in order to propagate. Surface waves moving across the ocean obviously require the water. Similarly, waves move along a plucked guitar string, across the surface of a struck drumhead, through Earth after an earthquake, and, indeed, in all materials acted upon by suitable forces. The speed of the waves depends on the properties of the medium and is derived relative to the medium. For example, the speed of sound waves in air, i.e., their absolute motion relative to still air, can be measured. The Doppler ef- fect for sound in air depends not only on the relative motion of the source and listener, but also on the motion of each relative to still air. Thus, it was natural for scientists of that time to expect the existence of some material like the ether to support the propa- gation of light and other electromagnetic waves and to expect that the absolute mo- tion of Earth through the ether should be detectable, despite the fact that the ether had not been observed previously. Michelson realized that although the effect of Earth’s motion on the results of any Albert A. Michelson, here “out-and–back” speed of light measurement, such as shown generically in Figure 1-5, playing pool in his later would be too small to measure directly, it should be possible to measure v2> c2 by a dif- years, made the first accurate measurement of ference measurement, using the interference property of the light waves as a sensitive the speed of light while an “clock.” The apparatus that he designed to make the measurement is called the instructor at the U.S. Naval Michelson interferometer. The purpose of the Michelson-Morley experiment was to Academy, where he had measure the speed of light relative to the interferometer (i.e., relative to Earth), thereby earlier been a cadet. [AIP detecting Earth’s motion through the ether and thus verifying the latter’s existence. To Emilio Segrè Visual Archives.] illustrate how the interferometer works and the reasoning behind the experiment, let us first describe an analogous situation set in more familiar surroundings. Light source Mirror c –v v c +v Observer A B L Figure 1-5 Light source, mirror, and observer are moving with speed v relative to the ether. According to classical theory, the speed of light c, relative to the ether, would be c  v relative to the observer for light moving from the source toward the mirror and c  v for light reflecting from the mirror back toward the source. 8 Chapter 1 Relativity I (a ) B Ground River v L 1 2 A C Ground L (b ) v c c2 – v2 c c2 – v2 v A →B B →A Figure 1-6 (a) The rowers both row at speed c in still water. (See Example 1-1.) The current in the river moves at speed v. Rower 1 goes from A to B and back to A, while rower 2 goes from A to C and back to A. (b) Rower 1 must point the bow upstream so that the sum of the velocity vectors c  v results in the boat moving from A directly to B. His speed relative to the banks (i.e., points A and B) is then (c2  v 2)1>2. The same is true on the return trip. EXAMPLE 1-1 A Boat Race Two equally matched rowers race each other over courses as shown in Figure 1-6a. Each oarsman rows at speed c in still water; the current in the river moves at speed v. Boat 1 goes from A to B, a distance L, and back. Boat 2 goes from A to C, also a distance L, and back. A, B, and C are marks on the riverbank. Which boat wins the race, or is it a tie? (Assume c  v.) SOLUTION The winner is, of course, the boat that makes the round trip in the shortest time, so to discover which boat wins, we compute the time for each. Using the classical velocity transformation (Equations 1-3), the speed of 1 relative to the ground is (c2  v 2)1>2, as shown in Figure 1-6b; thus the round-trip time t1 for boat 1 is L L 2L t1  tASB  tBSA    2c  v 2 2 2c  v 2 2 2c2  v 2 1-4 v 2 1/2 2L a1  2 b a1   Áb 2L2L 1 v2   艐 v2 c c c 2 c2 c 1 2 A c where we have used the binomial expansion. Boat 2 moves downstream at speed c  v relative to the ground and returns at c  v, also relative to the ground. The round-trip time t2 is thus L L 2Lc t2    2 cv cv c  v2 1-5 a1  2  Á b 2L 1 2L v2  艐 c v2 c c 1 c2  1-1 The Experimental Basis of Relativity 9 which, you may note, is the same result obtained in our discussion of the speed of light experiment in the Classical Concept Review. The difference ¢t between the round-trip times of the boats is then a1  2 b  a1  b 2L v2 2L 1 v2 Lv 2 ¢t  t2  t1 艐 艐 1-6 c c c 2 c2 c3 The quantity Lv 2>c3 is always positive; therefore, t2  t1 and rower 1 has the faster average speed and wins the race. The Results Michelson and Morley carried out the experiment in 1887, repeating with a much-improved interferometer an inconclusive experiment that Michelson alone had performed in 1881 in Potsdam. The path length L on the new interferom- eter (Figure 1-7) was about 11 meters, obtained by a series of multiple reflections. Michelson’s interferometer is shown schematically in Figure 1-8a. The field of view seen by the observer consists of parallel alternately bright and dark interference bands, called fringes, as illustrated in Figure 1-8b. The two light beams in the inter- ferometer are exactly analogous to the two boats in Example 1-1, and Earth’s motion through the ether was expected to introduce a time (phase) difference as given by Adjustable Unsilvered Light source mirror Silvered glass plate Mirrors glass plate Mirrors Mirrors Telescope 5 4 1 2 3 Figure 1-7 Drawing of Michelson-Morley apparatus used in their 1887 experiment. The optical parts were mounted on a 5 ft square sandstone slab, which was floated in mercury, thereby reducing the strains and vibrations during rotation that had affected the earlier experiments. Observations could be made in all directions by rotating the apparatus in the horizontal plane. [From R. S. Shankland, “The Michelson-Morley Experiment,” Copyright © November 1964 by Scientific American, Inc. All rights reserved.] 10 Chapter 1 Relativity I (a ) M´2 (b ) 1 Fringe width B M1 v 1 Rotation L Beam splitter Compensator M2 A C 2 Sodium light source L (diffuse) O Figure 1-8 Michelson interferometer. (a) Yellow light from the sodium source is divided into two beams by the second surface of the partially reflective beam splitter at A, at which point the two beams are exactly in phase. The beams travel along the mutually perpendicular paths 1 and 2, reflect from mirrors M1 and M2 , and return to A, where they recombine and are viewed by the observer. The compensator’s purpose is to make the two paths of equal optical length, so that the lengths L contain the same number of light waves, by making both beams pass through two thicknesses of glass before recombining. M2 is then tilted slightly so that it is not quite perpendicular to M1. Thus, the observer O sees M1 and M 2œ , the image of M2 formed by the partially reflecting second surface of the beam splitter, forming a thin wedge-shaped film of air between them. The interference of the two recombining beams depends on the number of waves in each path, which in turn depends on (1) the length of each path and (2) the speed of light (relative to the instrument) in each path. Regardless of the value of that speed, the wedge-shaped air film between M1 and M 2œ results in an increasing path length for beam 2 relative to beam 1, looking from left to right across the observer’s field of view; hence, the observer sees a series of parallel interference fringes as in (b), alternately yellow and black from constructive and destructive interference, respectively. Equation 1-6. Rotating the interferometer through 90° doubles the time difference and changes the phase, causing the fringe pattern to shift by an amount ¢N. An im- proved system for rotating the apparatus was used in which the massive stone slab on which the interferometer was mounted floated on a pool of mercury. This dampened vibrations and enabled the experimenters to rotate the interferometer without intro- ducing mechanical strains, both of which would cause changes in L and hence a shift in the fringes. Using a sodium light source with  590 nm and assuming v  30 km> s (i.e., Earth’s orbital speed), ¢N was expected to be about 0.4 of the width of a fringe, about 40 times the minimum shift (0.01 fringe) that the interferometer was capable of detecting.  1-2 Einstein’s Postulates 11 To Michelson’s immense disappointment and that of most scientists of the time, the expected shift in the fringes did not occur. Instead, the shift observed was only about 0.01 fringe, i.e., approximately the experimental uncertainty of the apparatus. With characteristic reserve, Michelson described the results thus:6 The actual displacement [of the fringes] was certainly less than the twentieth part [of 0.4 fringe], and probably less than the fortieth part. But since the displace- ment is proportional to the square of the velocity, the relative velocity of the earth and the ether is probably less than one-sixth the earth’s orbital velocity and cer- tainly less than one-fourth. Michelson and Morley had placed an upper limit on Earth’s motion relative to the ether of about 5 km> s. From this distance in time it is difficult for us to appreciate the devastating impact of this result. The then-accepted theory of light propagation could not be correct, and the ether as a favored frame of reference for Maxwell’s equa- tions was not tenable. The experiment was repeated by a number of people more than a dozen times under various conditions and with improved precision, and no shift has Michelson interferometers ever been found. In the most precise attempt, the upper limit on the relative velocity with arms as long as 4 km was lowered to 1.5 km> s by Georg Joos in 1930 using an interferometer with light are currently being used in paths much longer than Michelson’s. Recent, high-precision variations of the experi- the search for gravity waves. ment using laser beams have lowered the upper limit to 15 m> s. See Section 2-5. More generally, on the basis of this and other experiments, we must conclude that Maxwell’s equations are correct and that the speed of electromagnetic radiation is the same in all inertial reference systems independent of the motion of the source relative to the observer. This invariance of the speed of light between inertial reference frames means that there must be some relativity principle that applies to electromagnetism as well as to mechanics. That principle cannot be Newtonian relativity, which implies the dependence of the speed of light on the relative motion of the source and observer. It follows that the Galilean transformation of coordinates between inertial frames cannot be correct but must be replaced with a new coordinate transformation whose application preserves the invariance of the laws of electromagnetism. We then expect that the fundamental laws of mechanics, which were consistent with the old Galilean transformation, will require modification in order to be invariant under the new trans- formation. The theoretical derivation of that new transformation was a cornerstone of Einstein’s development of special relativity. More A more complete description of the Michelson-Morley experiment, its interpretation, and the results of very recent versions can be found on the home page: www.whfreeman.com/tiplermodernphysics5e. See also Figures 1-9 through 1-11 here, as well as Equations 1-7 through 1-10. 1-2 Einstein’s Postulates In 1905, at the age of 26, Albert Einstein published several papers, among which was one on the electrodynamics of moving bodies.11 In this paper, he postulated a more general principle of relativity that applied to the laws of both electrodynamics and mechanics. A consequence of this postulate is that absolute motion cannot be detected 12 Chapter 1 Relativity I by any experiment. We can then consider the Michelson apparatus and Earth to be at rest. No fringe shift is expected when the interferometer is rotated 90°, since all di- rections are equivalent. The null result of the Michelson-Morley experiment is there- fore to be expected. It should be pointed out that Einstein did not set out to explain the Michelson-Morley experiment. His theory arose from his considerations of the theory of electricity and magnetism and the unusual property of electromagnetic waves that they propagate in a vacuum. In his first paper, which contains the complete theory of special relativity, he made only a passing reference to the experimental at- tempts to detect Earth’s motion through the ether, and in later years he could not re- call whether he was aware of the details of the Michelson-Morley experiment before he published his theory. The theory of special relativity was derived from two postulates proposed by Einstein in his 1905 paper: Postulate 1. The laws of physics are the same in all inertial reference frames. Postulate 2. The speed of light in a vacuum is equal to the value c, independent of the motion of the source. Postulate 1 is an extension of the Newtonian principle of relativity to include all types of physical measurements (not just measurements in mechanics). It implies that no inertial system is preferred over any other; hence, absolute motion cannot be de- tected. Postulate 2 describes a common property of all waves. For example, the speed of sound waves does not depend on the motion of the sound source. When an ap- proaching car sounds its horn, the frequency heard increases according to the Doppler effect, but the speed of the waves traveling through the air does not depend on the speed of the car. The speed of the waves depends only on the properties of the air, such (a ) R1 as its temperature. The force of this postulate was to include light waves, for which v experiments had found no propagation medium, together with all other waves, whose speed was known to be independent of the speed of the source. Recent analysis of the R2 light curves of gamma-ray bursts that occur near the edge of the observable universe S have shown the speed of light to be independent of the speed of the source to a preci- sion of one part in 1020. (b ) R1 v Although each postulate seems quite reasonable, many of the implications of the v two together are surprising and seem to contradict common sense. One important im- R2 plication of these postulates is that every observer measures the same value for the S speed of light independent of the relative motion of the source and observer. Consider Figure 1-12 (a) Stationary a light source S and two observers R1 , at rest relative to S, and R2 , moving toward S with speed v, as shown in Figure 1-12a. The speed of light measured by R1 is c  3 108 m> s. What is the speed measured by R2? The answer is not c  v, as one light source S and a stationary observer R1 , with a second observer R2 moving toward would expect based on Newtonian relativity. By postulate 1, Figure 1-12a is equiva- the source with speed v. lent to Figure 1-12b, in which R2 is at rest and the source S and R1 are moving with (b) In the reference frame in speed v. That is, since absolute motion cannot be detected, it is not possible to say which the observer R2 is at which is really moving and which is at rest. By postulate 2, the speed of light from a rest, the light source S and moving source is independent of the motion of the source. Thus, looking at Figure observer R1 move to the right 1-12b, we see that R2 measures the speed of light to be c, just as R1 does. This result, with speed v. If absolute that all observers measure the same value c for the speed of light, is often considered motion cannot be detected, an alternative to Einstein’s second postulate. the two views are equivalent. Since the speed of light does This result contradicts our intuition. Our intuitive ideas about relative velocities not depend on the motion of are approximations that hold only when the speeds are very small compared with the the source, observer R2 speed of light. Even in an airplane moving at the speed of sound, it is not possible to measures the same value for measure the speed of light accurately enough to distinguish the difference between the that speed as observer R1. results c and c  v, where v is the speed of the plane. In order to make such a  1-2 Einstein’s Postulates 13 distinction, we must either move with a very great velocity (much greater than that of sound) or make extremely accurate measurements, as in the Michelson-Morley ex- periment, and when we do, we will find, as Einstein pointed out in his original rela- tivity paper, that the contradictions are “only apparently irreconcilable.” Events and Observers In considering the consequences of Einstein’s postulates in greater depth, i.e., in de- veloping the theory of special relativity, we need to be certain that meanings of some important terms are crystal clear. First, there is the concept of an event. A physical event is something that happens, like the closing of a door, a lightning strike, the col- lision of two particles, your birth, or the explosion of a star. Every event occurs at some point in space and at some instant in time, but it is very important to recognize that events are independent of the particular inertial reference frame that we might use to describe them. Events do not “belong” to any reference frame. Events are described by observers who do belong to particular inertial frames of reference. Observers could be people (as in Section 1-1), electronic instruments, or other suitable recorders, but for our discussions in special relativity we are going to be very specific. Strictly speaking, the observer will be an array of recording clocks lo- cated throughout the inertial reference system. It may be helpful for you to think of the observer as a person who goes around reading out the memories of the recording clocks or receives records that have been transmitted from distant clocks, but always keep in mind that in reporting events, such a person is strictly limited to summarizing the data collected from the clock memories. The travel time of light precludes him from including in his report distant events that he may have seen by eye! It is in this sense that we will be using the word observer in our discussions. Each inertial reference frame may be thought of as being formed by a cubic three- (Top) Albert Einstein in 1905 at the Bern, Switzerland, dimensional lattice made of identical measuring rods (e.g., meter sticks) with a patent office. [Hebrew recording clock at each intersection as illustrated in Figure 1-13. The clocks are all University of Jerusalem Albert identical, and we, of course, want them all to read the “same time” as one another at Einstein Archives, courtesy AIP any instant; i.e., they must be synchronized. There are many ways to accomplish syn- Emilio Segrè Visual Archives.] chronization of the clocks, but a very straightforward way, made possible by the sec- (Bottom) Clock tower and ond postulate, is to use one of the clocks in the lattice as a standard, or reference clock. electric trolley in Bern on For convenience we will also use the location of the reference clock in the lattice as Kramstrasse, the street on the coordinate origin for the reference frame. The reference clock is started with its which Einstein lived. If you indicator (hands, pointer, digital display) set at zero. At the instant it starts, it also are on the trolley moving sends out a flash of light that spreads out as a spherical wave in all directions. When away from the clock and look the flash from the reference clock reaches the lattice clocks 1 meter away (notice that back at it, the light you see must catch up with you. If in Figure 1-13 there are six of them, two of which are off the edges of the figure), we want their indicators to read the time required for light to travel 1 m ( 1> 299,792,458 s). you move at nearly the speed of light, the clock you see This can be done simply by having an observer at each clock set that time on the in- will be slow. In this, Einstein dicator and then having the flash from the reference clock start them as it passes. The saw a clue to the variability clocks 1 m from the origin now display the same time as the reference clock; i.e., they of time itself. [Underwood & are all synchronized. In a similar fashion, all of the clocks throughout the inertial Underwood/CORBIS.] frame can be synchronized since the distance of any clock from the reference clock can be calculated from the space coordinates of its position in the lattice and the initial setting of its indicator will be the corresponding travel time for the reference light flash. This procedure can be used to synchronize the clocks in any inertial frame, but it does not synchronize the clocks in reference frames that move with respect to one another. Indeed, as we shall see shortly, clocks in relatively moving frames cannot in general be synchronized with one another. 14 Chapter 1 Relativity I Figure 1-13 Inertial reference frame formed y from a lattice of measuring rods with a clock at each intersection. The clocks are all synchronized using a reference clock. In this diagram the measuring rods are shown to be 1 m long, but they could all be 1 cm, 1 m, or 1 km as required by the scale and precision of the measurements being considered. The three space dimensions are the clock positions. The fourth spacetime dimension, time, is shown by indicator readings on the clocks. Reference clock x z When an event occurs, its location and time are recorded instantly by the nearest clock. Suppose that an atom located at x  2 m, y  3 m, z  4 m in Figure 1-13 emits a tiny flash of light at t  21 s on the clock at that location. That event is recorded in space and in time or, as we will henceforth refer to it, in the spacetime coordinate sys- tem with the numbers (2,3,4,21). The observer may read out and analyze these data at his leisure, within the limits set by the information transmission time (i.e., the light travel time) from distant clocks. For example, the path of a particle moving through the lattice is revealed by analysis of the records showing the particle’s time of passage at each clock’s location. Distances between successive locations and the corresponding time dif- ferences make possible the determination of the particle’s velocity. Similar records of the spacetime coordinates of the particle’s path can, of course, also be made in any inertial frame moving relative to ours, but to compare the distances and time intervals measured in the two frames requires that we consider carefully the relativity of simultaneity. Relativity of Simultaneity Einstein’s postulates lead to a number of predictions about measurements made by ob- servers in inertial frames moving relative to one another that initially seem very strange, including some that appear paradoxical. Even so, these predictions have been experimentally verified; and nearly without exception, every paradox is resolved by an understanding of the relativity of simultaneity, which states that Two spatially separated events simultaneous in one reference frame are not, in general, simultaneous in another inertial frame moving relative to the first.  1-2 Einstein’s Postulates 15 A corollary to this is that Clocks synchronized in one reference frame are not, in general, synchronized in another inertial frame moving relative to the first. What do we mean by simultaneous events? Suppose two observers, both in the in- ertial frame S at different locations A and B, agree to explode bombs at time to (remember, we have synchronized all of the clocks in S). The clock at C, equidistant from A and B, will record the arrival of light from the explosions at the same instant, i.e., simultaneously. Other clocks in S will record the arrival of light from A or B first, de- pending on their locations, but after correcting for the time the light takes to reach each clock, the data recorded by each would lead an observer to conclude that the explosions were simultaneous. We will thus define two events to be simultaneous in an inertial reference frame if the light signals from the events reach an observer halfway between them at the same time as recorded by a clock at that location, called a local clock. Einstein’s Example To show that two events that are simultaneous in frame S are not simultaneous in another frame S moving relative to S, we use an example intro- duced by Einstein. A train is moving with speed v past a station platform. We have ob- servers located at A, B, and C at the front, back, and middle of the train. (We con- sider the train to be at rest in S and the platform in S.) We now suppose that the train and platform are struck by lightning at the front and back of the train and that the lightning bolts are simultaneous in the frame of the platform (S; Figure 1-14a). That is, an observer located at C halfway between positions A and B, where lightning strikes, observes the two flashes at the same time. It is convenient to suppose that the (a) S´ v B´ C´ A´ S B C A (b) S´ v B´ C´ A´ Figure 1-14 Lightning bolts strike the front and rear of the train, scorching S both the train and the platform, as the B C A train (frame S) moves past the platform (system S) at speed v. (a) The strikes are (c) v simultaneous in S, reaching the C S´ B´ C´ A´ observer located midway between the events at the same instant as recorded by S B C A the clock at C as shown in (c). In S the flash from the front of the train is recorded by the C clock, located (d) S´ v midway between the scorch marks on B´ C´ A´ the train, before that from the rear of the S train (b and d, respectively). Thus, the B C A C observer concludes that the strikes were not simultaneous. 16 Chapter 1 Relativity I lightning scorches both the train and the platform so that the events can be easily lo- cated in each reference frame. Since C is in the middle of the train, halfway between the places on the train that are scorched, the events are simultaneous in S only if the clock at C records the flashes at the same time. However, the clock at C records the flash from the front of the train before the flash from the back. In frame S, when the light from the front flash reaches the observer at C, the train has moved some dis- tance toward A, so that